Solving AB Beam Reactions with Hyperstatic System

[Zouatine]

hello everyone
1. Homework Statement
Problem:
An AB beam, loaded with a distributed load q (KN/m), the length of the beam is 2 m.
EI=constant

find the reactions in the beam ,we use y''=-(1/EI) *M(x)

Homework Equations

y''=-(1/EI) *M(x)[/B]

The Attempt at a Solution

first we have 6 reactions (Va,Ha,Ma,Vb,Hb,Mb) in the supports.
∑F/x=0→ Ha+Hb=0 , Ha=-Hb=0 -------------1
∑F/y=0→ Va+Vb-Q=0 → Va+Vb=Q→ Va+Vb=∫ ((e^x)+(e^-x))dx 0≤ x ≤2
Va+Vb= 7,25 KN -------------2
∑M/b=0 → 2*Va- Q*(center of gravity of the load q)-Ma+Mb=0
center of gravity of the load q = (∫∫x*ds)/(∫∫ds)
∫∫x*ds= ∫∫x*dx*dy 0≤y≤(e^x)+(e^-x) , 0≤x≤2
∫∫x*ds= ∫∫x*dx*dy = 9,25 KN*m
∫∫ds= 7,25 KN
center of gravity of the load q = (∫∫x*ds)/(∫∫ds)=1,28 m (2-1,28)= 0,72m
2*Va- Q*0,72-Ma+Mb=0
2*Va-Ma+Mb=5,22 KN m----------3
Differential equation:
y''=-(1/EI) *M(x)
M(x) = ?? 0≤x<2
my problem is how to find the moment equation?
thanx

 

[PhanthomJay]

hello everyone
1. Homework Statement
Problem:
An AB beam, loaded with a distributed load q (KN/m), the length of the beam is 2 m.
EI=constant
View attachment 236680
find the reactions in the beam ,we use y''=-(1/EI) *M(x)
2. Homework Equations
y''=-(1/EI) *M(x)
3. The Attempt at a Solution
first we have 6 reactions (Va,Ha,Ma,Vb,Hb,Mb) in the supports.
∑F/x=0→ Ha+Hb=0 , Ha=-Hb=0 -------------1
∑F/y=0→ Va+Vb-Q=0 → Va+Vb=Q→ Va+Vb=∫ ((e^x)+(e^-x))dx 0≤ x ≤2
Va+Vb= 7,25 KN -------------2
∑M/b=0 → 2*Va- Q*(center of gravity of the load q)-Ma+Mb=0
center of gravity of the load q = (∫∫x*ds)/(∫∫ds)
∫∫x*ds= ∫∫x*dx*dy 0≤y≤(e^x)+(e^-x) , 0≤x≤2
∫∫x*ds= ∫∫x*dx*dy = 9,25 KN*m
∫∫ds= 7,25 KN
center of gravity of the load q = (∫∫x*ds)/(∫∫ds)=1,28 m (2-1,28)= 0,72m
2*Va- Q*0,72-Ma+Mb=0
2*Va-Ma+Mb=5,22 KN m----------3
Differential equation:
y''=-(1/EI) *M(x)
M(x) = ?? 0≤x<2
my problem is how to find the moment equation?
thanx

i haven’t checked your calculus, but you seem to be on the right track. However, are you sure that both ends are fixed? If so, the problem is statically indeterminate to the second degree, and you have to resort to other equations besides the equilibrium equations, like deflection compatibility equations, virtual work, or other methods, which is a bit tedious.